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Properties

Label 2-1620-1620.1399-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.211 + 0.977i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 + 0.727i)2-s + (−0.396 + 0.918i)3-s + (−0.0581 + 0.998i)4-s + (−0.993 + 0.116i)5-s + (−0.939 + 0.342i)6-s + (−1.06 − 0.536i)7-s + (−0.766 + 0.642i)8-s + (−0.686 − 0.727i)9-s + (−0.766 − 0.642i)10-s + (−0.893 − 0.448i)12-s + (−0.342 − 1.14i)14-s + (0.286 − 0.957i)15-s + (−0.993 − 0.116i)16-s + (0.0581 − 0.998i)18-s + (−0.0581 − 0.998i)20-s + (0.914 − 0.767i)21-s + ⋯
L(s)  = 1  + (0.686 + 0.727i)2-s + (−0.396 + 0.918i)3-s + (−0.0581 + 0.998i)4-s + (−0.993 + 0.116i)5-s + (−0.939 + 0.342i)6-s + (−1.06 − 0.536i)7-s + (−0.766 + 0.642i)8-s + (−0.686 − 0.727i)9-s + (−0.766 − 0.642i)10-s + (−0.893 − 0.448i)12-s + (−0.342 − 1.14i)14-s + (0.286 − 0.957i)15-s + (−0.993 − 0.116i)16-s + (0.0581 − 0.998i)18-s + (−0.0581 − 0.998i)20-s + (0.914 − 0.767i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.211 + 0.977i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2606397603\)
\(L(\frac12)\) \(\approx\) \(0.2606397603\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.686 - 0.727i)T \)
3 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (0.993 - 0.116i)T \)
good7 \( 1 + (1.06 + 0.536i)T + (0.597 + 0.802i)T^{2} \)
11 \( 1 + (0.686 - 0.727i)T^{2} \)
13 \( 1 + (-0.893 - 0.448i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (1.59 - 0.802i)T + (0.597 - 0.802i)T^{2} \)
29 \( 1 + (-0.479 + 1.60i)T + (-0.835 - 0.549i)T^{2} \)
31 \( 1 + (-0.396 - 0.918i)T^{2} \)
37 \( 1 + (-0.766 + 0.642i)T^{2} \)
41 \( 1 + (1.33 - 1.41i)T + (-0.0581 - 0.998i)T^{2} \)
43 \( 1 + (0.914 - 1.22i)T + (-0.286 - 0.957i)T^{2} \)
47 \( 1 + (1.14 - 0.754i)T + (0.396 - 0.918i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.686 + 0.727i)T^{2} \)
61 \( 1 + (0.0460 + 0.790i)T + (-0.993 + 0.116i)T^{2} \)
67 \( 1 + (0.164 + 0.549i)T + (-0.835 + 0.549i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.0581 - 0.998i)T^{2} \)
83 \( 1 + (-1.33 - 1.41i)T + (-0.0581 + 0.998i)T^{2} \)
89 \( 1 + (1.05 - 0.882i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.973 - 0.230i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.936698137351505869923972256953, −9.539644525452779048907004222367, −8.234852406567006719701806135666, −7.85408211831137319034718610049, −6.55004516761659707343482861625, −6.33229121713818410964604144721, −5.11814972854710348772129027540, −4.28528409210505449534638454642, −3.66697043430972204314552287461, −2.98125789427096495595732549184, 0.16114586258492792363473214982, 1.82061152380822897364741018635, 2.95360529286570827331302676578, 3.72202316752530682804660587507, 4.88749473717156196530579512005, 5.69536047388336277949014506835, 6.58749950609473779818841847974, 7.08723584098129410453020669145, 8.357389040072005325725027764199, 8.904007042995966469716709480556

Graph of the $Z$-function along the critical line